Theoretical basis
2.11 Geographical terms
2.11.1 Haversine formula
The Haversine formula can calculate the distance between two points on a circle along the sur-face of the circle using the latitude and longitude provided [97]. Given the two points in latitude and longitude, the distanced can be calculated as the inverse of the haversineh. When calcu-lating the distance, the radiusr of the circle is important. The haversine is defined ash=d
r. d can be calculated as follows [78]:
• lat1= latitude of point 1
• lon1= longitude of point 1
• lat2= latitude of point 2
• lon2= longitude of point 2
• R= 6371e3; // metres
φis latitude, δis longitude. Ris earth’s radius (mean radius = 6,371km). δφis the change between latitudes while the∆δis change in between longitudes withφandδdescribed in radians.
a=sin2(∆φ
2 )+cosΦ1·cosΦ2·sin2(∆δ
2 ) (2.8)
Thereafter, we use the change of latitude and longitude into2.8, which gives usea, where a denotes the square of half the chord length between the points.
c=2·at an2(p
wherecfrom either2.9or2.10describes the angular distance in radians.
r esul t=R·c (2.11)
Which together gives us the distance between the two coordinates in metres (as the angular distance in radians is rescaled with the earth’s radiusR).
2.11.2 Earth’s radius
When calculating distances using the Haversine function on earth, it is crucial to consider that the earth’s radius is not constant. The shape of the earth is not that of a globe but that of an oblate spheroid. The radius varies from around 6378K mat the equator to 6357K mat the poles [94]. Due to the various in radius, the distance travelled by an object along the earth’s surface between two equally distant points might be different depending on the latitude of those points.
Therefore the radius of the earth needs to be calculated at the location that the calculation will take place with the formula2.11.2.
r(θ)=
s(a2·cosθ)2+(b2·sinθ)2
(a·cosθ)2+(b·sinθ)2 (2.12)
2.12 Hydrodynamics
Hydrodynamics is the science of the flow of fluid and how it affects its surroundings. In hydrody-namics, a fluid is viewed as its flow patterns; fluids flow at different speeds at different locations.
Sometimes the flow of fluids is parallel, but fluid flows are more often chaotic and frequently form whirlpools [42].
2.12.1 Forces on a body in water
Figure 2.2: Simulated vortex shedding, by Cesareo de La Rosa Siqueira [113].
2.12.1.1 Drag and inertia
The dampening primarily comes from drag and inertia. The drag forces can be described as friction between the object and the fluid. Inertia forces can be described as the forces related to the acceleration of the mass of the water that the body would displace. The drag and the inertial forces act on the body in a direction opposite to its movement [89].
Drag forces are dependent on the shape of the body, as well as its surface roughness. Inertia forces are dependent on the volume.
2.12.1.2 Turbulence: shapes and vortex shedding
Turbulence adds complexity to the problem. Turbulence is, in essence, the absence of laminar flow. The most significant effect of turbulence on an ROV comes from vortex shedding or sep-aration of flow caused by sharp edges, abrupt corners, or other non-aerodynamic features on the body [42]. These forces can have a self-amplifying effect if the separation of flow occurs at regular intervals between different features or to alternating sides of a shape where the flow can pass on both sides. This effect is called vortex shedding [22].
2.12.2 Stability and buoyancy
For a fully submerged rigid body, the center of gravity (CG) will always lay directly below the center of buoyancy(CB). If the center of gravity is shifted, the rigid object will rotate, so the statement above is fulfilled [63]. For a rigid body, this is shown in Figure2.3. Adding positive buoyancy to the center-top and more weight to the center-bottom, the stability of a rigid body will improve because more force has to be applied to rotate a rigid body [63].
Figure 2.3: A fully submerged rigid body [63]
2.12.3 Hydrodynamics and Towed vehicles
The Hydrodynamics of a Towed ROV can often be complex, but the principles are simple. A Towed ROV needs to be stable in the water, with minimal unintended changes in pitch, roll, yaw or position. The combined hydrodynamic and hydrostatic forces acting on the body, as well as the orientation of the center of buoyancy and the center of mass, determine the ROV’s stability [29]. The forces are determined by the ROV’s speed and shape, fluid properties, buoyancy, and mass. The hydrodynamic and hydrostatics are dominant at high speed, while at low speed, buoyancy and gravity play a more critical role [31]. When an object moves relative to the water, the object’s shape will create turbulence, but sharper edges will create more vector shedding and therefore have a more pronounced effect on the body, as seen in Figure2.4.
Figure 2.4: A Towed object disturbs the water flow around it
2.12.4 Hydrofoil
Foil is a wing or flipper that produces both a pressure and a sucking force on the wing as it moves through a fluid [72]. The laminar flow around the wing creates these forces. As seen in Figure2.5 the flow around the wing is compressed on one side, while on the other side creating a sucking force. These disturbances in flow around the wing create an imbalance in pressure above and below the wing, resulting in a force in a different direction compared to the flow. This force is called lift.
Figure 2.5: Waterflow is displaced around a wing, creating pressure and sucking forces.
2.12.4.1 Stall
In hydrodynamics, the stall is the loss of lift force due to the wing’s angle of attack (AOT) [72].
If the AOT is too steep, the turbulence created by the wing disturbs the standard flow patterns.
In this case, the wing loses its pressure vs sucking imbalance and instead operates more like a brake; This will still generate a force on the wing due to the pressure of water being displaced, but this force is often smaller than the forces created by a hydrofoil [80].